Singular value decomposition and spectral analysis (original) (raw)
Singular-value decomposition approach to time series modelling
IEE Proceedings F Communications, Radar and Signal Processing, 1983
In various signal processing applications, as exemplified by spectral analysis, deconvolution and adaptive filtering, the parameters of a linear recursive model are to be selected so that the model is 'most' representative of a given set of time series observations. For many of these applications, the parameters are known to satisfy a theoretical recursive relationship involving the time series' autocorrelation lags. Conceptually, one may then use this recursive relationship, with appropriate autocorrelation lag estimates substituted, to effect estimates for the operator's parameters. A procedure for carrying out this parameter estimation is given which makes use of the singular-value decomposition (SVD) of an extended-order autocorrelation matrix associated with the given time series. Unlike other SVD modelling methods, however, the approach developed does not require a full-order SVD determination. Only a small subset of the matrix's singular values and associated characteristic vectors need be computed. This feature can significantly alleviate an otherwise overwhelming computational burden that is necessitated when generating a full-order SVD. Furthermore, the modelling performance of this new method has been found empirically to excel that of a near maximum-likelihood SVD method as well as several other more traditional modelling methods. 'The symbol [«,, n 2 ] denotes the set of integers satisfying n x < n < n2 while [ n,, °°) specifies the set of integers satisfying n > n,.
On the use of Singular Spectrum Analysis
Singular Spectrum Analysis (SSA) or Singular Value Decomposition (SVD) are often used to de-noise univariate time series or to study their spectral profile. Both techniques rely on the eigendecomposition of the correlation matrix estimated after embedding the signal into its delayed coordinates. In this work we show that the eigenvectors can be used to calculate the coefficients of a set of filters which form a filter bank. The properties of these filters are derived. In particular we show that their outputs can be grouped according to their frequency response. Furthermore, the frequency at the maximum of each frequency response and the corresponding eigenvalue can provide a power spectrum estimation of the time series. Two different applications illustrate how both characteristics can be applied to analyze wideband signals in order to achieve narrow-band signals or to infer their frequency occupation.
WIREs Computational Statistics
Singular spectrum analysis (SSA), starting from the second half of XX century, has been a rapidly developing method of time series analysis. Since it can be called principal component analysis for time series, SSA will definitely be a standard method in time series analysis and signal processing in the future. Moreover, the problems solved by SSA are considerably wider than that for PCA. In particular, the problems of frequency estimation, forecasting and missing values imputation can be solved within the framework of SSA. The idea of SSA came from different scientific communities such as time series analysis (Karhunen-Loeve decomposition), signal processing (low-rank approximation and frequency estimation) and multivariate data analysis (principal component analysis). Also, depending on the area of applications, different viewpoints on the same algorithms, choice of parameters, and methodology as a whole are considered. Thus, the aim of the paper is to describe and compare different viewpoints on SSA and its modifications and extensions to give people from different scientific communities the possibility to be aware with potentially new aspects of the method.
Signal decomposition and time–frequency representation using iterative singular spectrum analysis
Geophysical Journal International, 2019
The application of the singular value decomposition method (SVD) for filtering of seismic data has become common in recent decades, as it promotes significant improvements of the signalto-noise ratio, highlighting reflections in seismograms. One particular way to apply SVD in a single (or multivariate) time-series is the singular spectrum analysis (SSA) method, normally applied on constant-frequency slices in one or many spatial dimensions. We demonstrate that SSA method applied in the time domain corresponds to filtering the time-series with a symmetric zero-phase filters, which are the autocorrelations of the eigenvectors of the data covariance matrix, preserving the phase of the original data. In this paper, we explore the SSA method in the time domain, and we propose a new recursive-iterative SSA (RI-SSA) algorithm, which uses only the first eigenvector of the data covariance matrix to decompose a discrete time-series into signal components. From the analytic signal of each component we compute a time-frequency representation. By interpretation of the time signals and their time-frequency representations, groups with similar features are summed to produce a smaller number of signal components. The resulting RI-SSA signal decomposition is exact and phasepreserving, but non-unique. Applications to land seismic data for ground-roll removal and to two synthetic signals for time-frequency analysis give good results.
The Sliding Singular Spectrum Analysis: a Data-Driven Non-Stationary Signal Decomposition Tool
IEEE Transactions on Signal Processing
Singular Spectrum Analysis (SSA) is a signal decomposition technique that aims at expanding signals into interpretable and physically meaningful components (e.g. sinusoids, noise, etc.). This article presents new theoretical and practical results about the separability of the SSA and introduces a new method called sliding SSA. First, the SSA is combined with an unsupervised classification algorithm to provide a fully automatic data-driven component extraction method for which we investigate the limitations for components separation in a theoretical study. Second, the detailed automatic SSA method is used to design an approach based on a sliding analysis window which provides better results than the classical SSA method when analyzing non-stationary signals with a time-varying number of components. Finally, the proposed sliding SSA method is compared to the Empirical Mode Decomposition (EMD) and to the synchrosqueezed Short-Time Fourier Transform (STFT), applied on both synthetic and real-world signals.
Estimating the parameters of exponentially damped sinusoids and pole-zero modeling in noise
Acoustics, Speech and Signal …, 1982
ARtract-We have presented techniques [ 11 -[ 6 ] based on linear prediction (LP) and singular value decomposition (SVD) for accurate estimation of closely spaced frequencies of sinusoidal signals in noise. In this note we extend these techniques to estimate the parameters of exponentially damped sinusoidal signals in noise. The estimation procedure presented here makes use of "backward prediction" in addition to SVD.
IEEE Transactions on Signal Processing, 1993
Recently, Wax and Kailath developed information theoretic criteria for detection of the number of signals received by a sensor array. More recently, Fuchs developed a criterion, based on the perturbation analysis of the data autocorrelation matrix, for detecting the number of sinusoids. In this paper, following the information theoretic approach to model selection, we first develop criteria for detection of the number of dampedlundamped sinusoids. These criteria are matched to the singular value decomposition (SVD) based methods, such as modified forwardlhackward and forwardbackward linear prediction, so well that the extra computations needed over and above those required for computing the SVD are marginal. Next, we develop an analytical framework for analyzing the performance of these criteria, following the assumptions made by Wang and Kaveh and the corrections given by Zhang et al. In the development of the analysis, we make some approximations which become better for large signal-tonoise ratio. Simulations are used to verify the usefulness of the analysis, and to compare the performance of our method with that of Fuchs.
Signal Identification in Singular Spectrum Analysis
Australian & New Zealand Journal of Statistics, 2016
This paper provides an information theoretic analysis of the signal identification problem in singular spectrum analysis. We present a signal-plus-noise model based on the Karhunen-Loève expansion and use this model to motivate the construction of a minimum description length criterion that can be employed to identify the dimension (rank) of the signal component. We show that under very general regularity conditions the criterion will identify the true signal dimension with probability one as the sample size increases. A by-product of this analysis is a procedure for selecting a window length consistent with the Whitney embedding theorem. The upshot is a modeling strategy that results in a specification that yields a signal-noise reconstruction that minimises mean squared reconstruction error. Empirical results obtained using simulated and real world data series indicate that theoretical properties presented in the paper are reflected in observed behaviour, even in relatively small samples, and that the minimum description length modeling strategy provides the practitioner with an effective addition to the SSA tool box.
A Review on Singular Spectrum Analysis
2022 IEEE International Conference on Current Development in Engineering and Technology (CCET)
Singular Spectrum Analysis (SSA), a relatively new but effective approach in time series analysis, has been devised and widely used in various of practical problems in the recent years. It is regarded as PCA for time series however has huge advantages over it. SSA will surely become a principal time series analysis method in the future. The aim of this paper is to aware scientific communities with the knowledge of various advances and extensions of SSA. This paper begins with brief history of SSA. The paper also discusses the Basic SSA algorithm and its stages. Here, in this paper, recent advances in the conceptual and algorithmic aspects of the SSA are studied upon and reviewed. These include the recent developments and extension of the Basic SSA. This paper also compares the Basic SSA with other classical methods, their similarities and differences from one and other.